We address the problem of computing simulation relations over tree
automata. In particular, we consider downward and upward simulations on
tree automata, which are, loosely speaking, analogous to forward and
backward relations over word automata. We provide simple and efficient
algorithms for computing these relations based on a reduction to the
problem of computing simulations on labelled transition systems.
Furthermore, we show that downward and upward relations can be combined
to get relations compatible with the tree language equivalence, which
can subsequently be used for an efficient size reduction of
nondeterministic tree automata. This is of a very high interest, for
instance, for symbolic verification methods such as regular model
checking, which use tree automata to represent infinite sets of
reachable configurations. We provide experimental results showing the
efficiency of our algorithms on examples of tree automata taken from
regular model checking computations.